Emergence of HIV-1 Drug Resistance During Antiretroviral ...fengz/pub/bmb07_hiv.pdf · stages of...

Bulletin of Mathematical Biology (2007)DOI 10.1007/s11538-007-9203-3

O R I G I NA L A RT I C L E

Emergence of HIV-1 Drug Resistance During AntiretroviralTreatment

Libin Ronga, Zhilan Fenga, Alan S. Perelsonb,∗

aDepartment of Mathematics, Purdue University, West Lafayette, IN 47907, USAbTheoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos,NM 87545, USA

Received: 15 August 2006 / Accepted: 9 February 2007© Society for Mathematical Biology 2007

Abstract Treating HIV-infected patients with a combination of several antiretroviraldrugs usually contributes to a substantial decline in viral load and an increase in CD4+

T cells. However, continuing viral replication in the presence of drug therapy can lead tothe emergence of drug-resistant virus variants, which subsequently results in incompleteviral suppression and a greater risk of disease progression. In this paper, we use a simplemathematical model to study the mechanism of the emergence of drug resistance duringtherapy. The model includes two viral strains: wild-type and drug-resistant. The wild-typestrain can mutate and become drug-resistant during the process of reverse transcription.The reproductive ratio R0 for each strain is obtained and stability results of the steadystates are given. We show that drug-resistant virus is more likely to arise when, in thepresence of antiretroviral treatment, the reproductive ratios of both strains are close. Thewild-type virus can be suppressed even when the reproductive ratio of this strain is greaterthan 1. A pharmacokinetic model including blood and cell compartments is employed toestimate the drug efficacies of both the wild-type and the drug-resistant strains. We inves-tigate how time-varying drug efficacy (due to the drug dosing schedule and suboptimaladherence) affects the antiviral response, particularly the emergence of drug resistance.Simulation results suggest that perfect adherence to regimen protocol will well suppressthe viral load of the wild-type strain while drug-resistant variants develop slowly. How-ever, intermediate levels of adherence may result in the dominance of the drug-resistantvirus several months after the initiation of therapy. When more doses of drugs are missed,the failure of suppression of the wild-type virus will be observed, accompanied by a rela-tively slow increase in the drug-resistant viral load.

Keywords HIV-1 · Antiretroviral therapy · Adherence · Drug resistance · Mutation ·Viral kinetics

∗Corresponding author.E-mail address: [email protected] (Alan S. Perelson).

L. Rong et al.

1. Introduction

Currently there are four classes of antiretroviral (ARV) drugs available in the treat-ment of HIV-1-infected patients: (1) nucleoside/nucleotide reverse transcriptase inhibitors(NRTIs), (2) non-nucleoside reverse transcriptase inhibitors (NNRTIs), (3) protease in-hibitors (PIs), and (4) entry/fusion inhibitors (EIs). Each class of drugs targets differentstages of the viral lifecycle. Antiretroviral therapy (ART) using a combination of three ormore drugs from two or more classes (for example, two NRTIs combined with either a PIor an NNRTI) has proved to be extremely effective in suppressing the plasma viral loadsof most HIV-1-infected patients below the limit of viral detection (50 RNA copies ml−1)by standard assay to date (Collier et al., 1996). However, drug treatment often fails toachieve complete viral suppression to below the limit of detection (virological failure)because of many host and viral factors, for example, nonadherence to the treatment pro-tocol, deleterious side effects, poor drug absorption, or the emergence of drug-resistantvirus (Deeks, 2003).

The question of why drug-resistant strains of HIV emerge during therapy is of greatinterest. Considerable progress has been made in the genotypic and phenotypic charac-terization of drug-resistant virus variants. The main phenotype of mutant strains is thereduction of drug susceptibility in the presence of therapy. The appearance of HIV drug re-sistance was first documented with some nucleoside inhibitors (for example, AZT Larderet al., 1989) that inhibit viral replication by virtue of their activities as DNA chain ter-minators. Most of our knowledge regarding the resistance to nucleoside analogues comesfrom the genotypic analysis of HIV isolates from patients receiving prolonged therapywith these drugs. Mutations at several codons of reverse transcriptase have been as-sociated with the resistant phenotype (De Jong et al., 1996; Larder and Kemp, 1989;Richman, 1992). Mutations detected during the clinical use of nevirapine (a NNRTI) haveconfirmed that as few as one amino acid change can generate resistance in vivo duringtherapy (Havlir et al., 1996; Richman et al., 1994). Exploring the association betweengenotypic and phenotypic characteristics will provide information that can be used tounderstand the emergence of drug resistance and better predict treatment outcomes.

Insights into HIV drug resistance have been obtained from mathematical modeling ofantiretroviral responses and the evolution of mutant virus. Kirschner and Webb (1997)studied drug resistance during treatment of HIV infection with a single drug and com-pared the treatment outcomes when drug therapy was initiated at different CD4+ T celllevels. McLean and Nowak (1992) showed that the competition between drug-resistantand wild-type strains determines which type of virus will eventually dominate the viruspopulation during the course of AZT treatment. Ribeiro et al. (1998) calculated the fre-quency of drug-resistant mutant virus before the initiation of therapy, and suggested thatdrug resistant virus can preexist in patients and then be selected when therapy is started.Nowak et al. (1997) discussed a two-strain model and compared the model results withdata on drug resistance development in patients treated with nevirapine. The role of an im-mune response in the emergence of drug resistance was investigated in (Shiri et al., 2005;Wodarz and Lloyd, 2004). Clinical data on the evolution of drug-resistant mutants fortwo RT inhibitors, lamivudine and zidovudine, was explained in detail by a mathematicalmodel that incorporated empirical evidence on the mutation frequency of various HIV-1drug resistance mutations (Stilianakis et al., 1997). Murray and Perelson (2005) showedhow the quasi-species nature of HIV can influence the development of resistance to AZT

Emergence of HIV-1 Drug Resistance During Antiretroviral

and the maintenance of drug resistance clones after cessation of therapy. Kepler and Perel-son (1998) showed how drug concentration heterogeneity facilitates the evolution of drugresistance. Smith and Wahl (2005) used an impulsive differential equation to model drugbehavior and classified three treatment regimens according to whether the drug level islow, intermediate or high. Their models predicted that drug resistance might arise at bothintermediate and high drug concentrations, whereas at low drug levels resistance wouldnot emerge.

In the clinic, HIV resistance can result from the transmission of drug-resistant mutantsto susceptible individuals (Blower et al., 2001) or from the acquisition of mutations gen-erated during treatment. It is important to distinguish between scenarios in which drug-resistant virus preexists before the onset of therapy or in which they are produced byresidual virus replication during the course of ART, because each process requires dif-ferent drug regimens to maximize the clinical benefits (Bonhoeffer and Nowak, 1997).The calculation of probabilities of both processes suggests that under a wide range ofconditions, treatment failure is most likely due to the preexistence of drug-resistant virusbefore therapy (Ribeiro and Bonhoeffer, 2000). Provided that drug-resistant virus preex-ists, Bonhoeffer and Nowak (1997) showed that a more efficient therapy would lead toa greater initial reduction of virus load, but would also cause a faster rise of resistancemutations.

Our primary goal in this paper is to investigate analytically the mechanisms underly-ing the emergence of drug-resistant variants during ART. We use a simple mathemat-ical model to study the effect of ARV drugs on the evolution of drug-resistant HIVmutants. Although HIV resistance is not an all-or-nothing phenomenon and generallymutations accumulate to provide more resistance to drug therapy (Clavel and Hance,2004), even a single mutation can confer a significant degree of resistance to a drugor an entire class of drugs (Bonhoeffer and Nowak, 1997; Larder and Kemp, 1989;Mugavero and Hicks, 2004; St Clair et al., 1991). For example, the M184V muta-tion in reverse transcriptase can result in complete resistance to lamivudine (Clavel andHance, 2004). In this paper, we assume that the drug-sensitive and resistant strainsdiffer by a single mutation. The model can be extended to include multiple resis-tant strains in which two or more point mutations are required (Ribeiro et al., 1998;Stilianakis et al., 1997). We will also discuss the evolution of resistant strains that requiremultiple mutations in the final section with the present model.

Adherence to prescribed ART is recognized as an essential principle in HIV treat-ment. Much evidence shows that suboptimal adherence is associated with a high riskof developing clinically significant HIV drug resistance (Bangsberg et al., 2001; Fried-land and Williams, 1999; Sethi et al., 2003; Wahl and Nowak, 2000). Richman (1996)postulated that the relationship between drug resistance and antiretroviral activity was a“bell-shaped” curve—that is, drug resistance is more likely to appear with moderate lev-els of adherence than with perfect or low levels of adherence to highly potent treatment.With perfect adherence there might be little viral replication, whereas with poor adher-ence there might be insufficient drug pressure to select resistant variants. A small num-ber of mathematical models have considered the effects of nonperfect adherence to drugregimens (Ferguson et al., 2005; Huang et al., 2003; Phillips et al., 2001; Smith, 2006;Tchetgen et al., 2001; Wahl and Nowak, 2000; Wu et al., 2006) and an overview can befound in Heffernan and Wahl (2005). Wahl and Nowak (2000) analyzed the outcome of

L. Rong et al.

therapy as a function of the degree of adherence to drug regimen and determined the con-ditions under which a resistant strain dominates. Wu et al. (2006) analyzed the effects ofpharmacokinetics and adherence on treatment outcome. Phillips et al. (2001) used a sto-chastic model to study the consequence for resistance development of different drug usepatterns. Smith (2006) also dealt with adherence and determined how many doses may bemissed before HIV treatment is adversely affected by the emergence of drug resistance.In Smith (2006), the dynamics of drug are modeled by impulsive differential equations.Thus, the author assumed that drug concentrations instantly increase after a dose is taken.However, this is not realistic since it does not account for the time needed for drug tobe absorbed and then transported and processed into an active form intracellularly (Dixitand Perelson, 2004). In this paper, we adopt a pharmacokinetic (PK) model developed re-cently by Dixit and Perelson (2004) to estimate the drug efficacies of tenofovir disoproxilfumarate (NRTI) and ritonavir (PI) when perfect adherence or suboptimal adherence isfollowed. The PK model considers the absorption of drug from the gut into the blood andthen its transport into the cell. For the case of tenofovir, the model also considers that themolecule needs to be phosphorylated intracellularly in order to become an active NRTI.Drug efficacies for both wild-type and drug-resistant strains are carefully estimated. Westudy how time-varying drug efficacies, due both to dosing regimens and nonadherence,affect the antiviral responses. We provide analytical and numerical results of the presentmodel and compare the resistance evolution in the absence/presence of therapy, with per-fect/suboptimal adherence to the treatment protocol.

2. The pretreatment two-strain model

The following model has been widely adopted to model the plasma viral load in HIVinfected patients (Nowak and May, 2000; Perelson et al., 1996):

d

dtT (t) = λ − dT − kV T ,

d

dtT ∗(t) = kT V − δT ∗,

d

dtV (t) = NδT ∗ − cV,

(1)

where T (t), T ∗(t) and V (t) denote the concentrations of uninfected target T cells, pro-ductively infected cells, and free virus at time t , respectively. The parameter λ representsthe recruitment rate of uninfected T cells, d is the per capita death rate of uninfected cells,k is the rate constant at which uninfected cells are infected by free virus. δ is the per capitadeath rate of infected cells, N (burst size) is the total number of virus particles releasedby a productively infected cell over its lifespan and c is the clearance rate of virus.

Significant insights into virus dynamics have been derived by estimating the kineticparameters δ and c when antiretroviral drug is used to perturb the pretreatment steadystates in HIV infected patients (Perelson et al., 1996, 1997; Wei et al., 1995). The stabilityof the steady states of model (1) is determined entirely by the basic reproductive ratio R0.


R0 is usually defined as the number of secondary infections in a healthy host caused by asingle infected cell, and is given by

R0 = kλN

dc.

The infection-free steady state is globally attracting if R0 < 1, while the infected steadystate is globally asymptotically stable when R0 > 1 (De Leenheer and Smith, 2003).

2.1. Two-strain mathematical model

HIV-1 replicates at an exceptionally high rate in untreated patients, with more than 1010

virus particles produced per day in mid-stage HIV-1 infected patients (Perelson et al.,1996). The process by which the HIV-1 RNA genome is reverse transcribed into DNAis highly error prone, and the probability that mutations occur is quite high (the averagenumber of changes per genome is 0.3 per replication cycle, which implies that after re-verse transcription about 22% of infected cells should carry proviral genomes with onemutation Perelson et al., 1997; Perelson and Nelson, 2002). As a single mutation or a num-ber of mutation combinations can result in drug resistance, there is a reasonable chancethat drug-resistant variants of HIV preexist even before the initiation of therapy (Ribeiroand Bonhoeffer, 2000). Here we use a mathematical model including both wild-type, i.e.,drug sensitive, and drug resistant strains to study the viral load of each strain in the ab-sence of ARV drugs. The model can be described by the following system of ordinarydifferential equations:

d

dtT (t) = λ − dT − ksVsT − krVrT ,

d

dtTs(t) = (1 − u)ksVsT − δTs,

d

dtVs(t) = NsδTs − cVs,

d

dtTr(t) = uksVsT + krVrT − δTr ,

d

dtVr(t) = NrδTr − cVr,

(2)

where Ts(t) is the concentration of cells productively infected by drug sensitive virus;Tr(t) is the concentration of cells productively infected by drug-resistant virus; Vs(t) andVr(t) represent the respective concentrations of drug sensitive and drug-resistant virus.ks and kr represent the rate constants at which uninfected cells are infected by drug sen-sitive and drug-resistant virus, respectively. u (0 ≤ u < 1) is the rate at which cells in-fected by the drug sensitive virus mutate and become drug-resistant during the processof reverse transcription of viral RNA into proviral DNA. Both types of infected cellsare assumed to have the same death rate δ. We suppose that the drug sensitive and re-sistant strains differ in their burst sizes, Ns and Nr , while they have the same virionclearance rate c. It should be noted that the backward mutation from drug-resistant

L. Rong et al.

to drug-sensitive strain is neglected since the wild-type virus dominates the popula-tion before the initiation of therapy (Bonhoeffer and Nowak, 1997; Nowak et al., 1997;Shiri et al., 2005). We discuss our assumptions about parameters and their possible effectson our results in Section 5.

Several two-strain models have been studied in the literature (Bonhoeffer et al., 1997;Bonhoeffer and Nowak, 1997; McLean and Nowak, 1992; Nowak et al., 1997; Ribeiroet al., 1998). However, many of them focus on the question of whether drug-resistantmutants preexist before the therapy and are selected in the presence of drugs or if theyare produced only in the course of treatment. Here we aim to study in depth the virusdynamics as well as the development of drug-resistant strains. Particularly, we attempt toaddress the following questions: Under what condition does the wild-type strain dominatein the absence of therapy? How does the drug-resistant strain develop when ARV drugsare used? Is it possible to eradicate both strains of virus? How soon will resistance appearif a small number of drug doses is missed? What is the situation if more doses are missed?

2.2. Model parameters

We use the following model parameters in all the simulations in this paper. An uncertaintyand sensitivity analysis is carried out to study the influence of the uncertainty of theseparameters on the outcome variable in Section 2.4.

Within uninfected individuals the average density of CD4+ T cells remains relativelyconstant at about 106 cells per milliliter (Bofill et al., 1992; Sedaghat and Siliciano, 2004).Here we assume all these cells are possible targets of infection, i.e., T0 = 106 ml−1. Thedeath rate of uninfected cells is chosen to be d = 0.01 day−1 (Mohri et al., 1998). Beforeinfection, the density of target cells is at equilibrium, thus λ = dT0 = 104 ml−1 day−1.The rate constant for target cells becoming infected by drug-sensitive virus, ks , is not wellknown and assumed to be ks = 2.4 × 10−8 ml day−1 (Perelson et al., 1993). The estimateof burst size of wild-type strain, Ns , varies from 100 to a few thousands (Haase et al.,1996; Hockett et al., 1999) and possibly could be significantly larger (Yuan Chen et al.,2007). Here, as an example, we choose Ns = 3000.

Since mutant strains are often associated with the changes of highly conserved amino-acid residues that are believed to be important for enzyme function, many resistant mu-tants display some extent of resistance-associated loss of fitness when compared withdrug-sensitive strains (Clavel et al., 2000). Recent evidence also shows that the emer-gence of drug resistance reduces the inherent replicative capacity of resistant strains al-though it increases the ability of HIV to replicate in the presence of drugs (Barbour et al.,2002). However, there have not been direct experimental measurements to date determin-ing whether the replicative defect is due to impaired infectivity of HIV or reduced viralproduction. Thus, in this paper, we assume that both the infection rate and burst size ofresistant strain are less than those of the wild-type strain, i.e., kr < ks and Nr < Ns , ashas been done previously (Snedecor, 2003). We choose kr = 2.0 × 10−8 ml day−1 andNr = 2000 in our numerical simulation.

The death rate of infected cells, δ, and the clearance rate of virus, c, are chosen tobe our current best estimates: δ = 1 day−1 (Markowitz et al., 2003) and c = 23 day−1

(Ramratnam et al., 1999). The mutation rate from the wild-type strain to a drug-resistantstrain is u = 3 × 10−5 (Mansky and Temin, 1995). This mutation rate applies only whenthe wild-type and the mutant strains differ by a single point mutation. In the case of


Table 1 Parameter definitions and values used in numerical simulations

Parameter Value Description Reference

λ 104 ml−1 day−1 Recruitment rate of uninfectedcells

See text

d 0.01 day−1 Death rate of uninfected cells Mohri et al. (1998)ks 2.4 × 10−8 ml day−1 Infection rate of target cells by

wild-type virusPerelson et al. (1993)

kr 2.0 × 10−8 ml day−1 Infection rate of target cells bydrug-resistant virus

See text

u 3 × 10−5 Mutation rate from sensitive strainto resistant strain

Mansky and Temin (1995)

δ 1 day−1 Death rate of infected cells Markowitz et al. (2003)Ns 3000 Burst size of drug-sensitive strain See textNr 2000 Burst size of drug-resistant strain See textc 23 day−1 Clearance rate of free virus Ramratnam et al. (1999)εsRT

Varied Efficacy of RTIs for wild type See text

εrRT

Varied Efficacy of RTIs for mutants See text

εsP I

Varied Efficacy of PIs for wild type See text

εrP I

Varied Efficacy of PIs for mutants See text

εs Varied Overall drug efficacy for wild type See textεr Varied Overall drug efficacy for mutants See textα Varied Resistance level of mutant strain See text

two or more point mutations, the probability of mutation directly from the wild-type toa resistant strain will be much smaller (an approximation is un where n is the numberof point mutations they differ by). We will discuss the evolution of drug-resistant strainswith multiple mutations later. A summary of the parameters used in this paper and theirvalues adopted in our simulations is given in Table 1.

2.3. Analysis of the pretreatment model

Model (2) has three possible steady states. The existence and stability of these steadystates are considered and shown to be dependent on the magnitudes of the reproductiveratios of both strains as well as the mutation rate.

2.3.1. Steady states of model (2)Let E = (T , Ts , Vs , Tr , Vr ) denote a constant solution (steady state) of model (2). Thereare three possible steady states: the infection-free steady state

E0 =(

λ

d,0,0,0,0

), (3)

the boundary steady state (only the drug-resistant strain is present)

Er =(

c

krNr

,0,0, (Rr − 1)dc

krNrδ, (Rr − 1)

d

kr

), (4)

L. Rong et al.

and the interior steady state (coexistence of both the wild-type and the drug-resistantstrains)

Ec =(

c

(1 − u)ksNs

, Ts,Nsδ

cTs, Tr ,

Nrδ

cTr

), (5)

where

Ts = [(1 − u)σ − 1][(1 − u)Rs − 1]λ(σ − 1)(1 − u)Rsδ

> 0,

Tr = [(1 − u)Rs − 1]uσλ

(σ − 1)(1 − u)Rsδ> 0,

σ = Rs/Rr , and Rs and Rr are the basic reproductive ratios of the wild-type strain andthe drug-resistant strain, respectively, and are given by

Rs = (ksNsλ)/(dc), Rr = (krNrλ)/(dc). (6)

Notice that in the special case u = 0 (i.e., there is no mutation), the interior steadystate Ec reduces to another boundary steady state Es (at which only the wild-type strainis present):

Es =(

c

ksNs

, (Rs − 1)dc

ksNsδ, (Rs − 1)

d

ks

,0,0

), (7)

and the other steady states (E0 and Er ) remain the same.It is clear that Er exists if and only if Rr > 1, and Ec exists if and only if Rs >

1/(1 − u) and σ > 1/(1 − u). We will show that these existence conditions also providethreshold conditions for the stability of these steady states.

From the expressions of the infected steady states (Er and Ec), we observe that drug-resistant mutants will always be present as long as mutation from the wild-type to themutant strain is possible (i.e., u �= 0). The fact that the resistant virus can completelyout compete the wild-type strain is due to the assumption that mutation only occurs inone direction from the wild-type to the resistant. If we also include back mutation in themodel, then the resistant strain can not exist alone although it can out-compete the wild-type strain. Which strain will dominate the virus population depends on the competitionof these two strains.

From the assumptions kr < ks and Nr < Ns , we have Rs > Rr > 1 in the absenceof treatment (with the parameters given in Table 1). Hence, σ is greater than 1 but notclose to 1. Therefore, from the formula for Vr (see (5)) we know that the viral load of theresistant strain is very low (e.g., it is below the detection limit when the parameter valuesin Table 1 are used, see Fig. 1) due to the low mutation rate u despite the coexistence ofboth strains. The more point mutations needed to confer drug resistance for the mutantstrain, the lower the viral load of the resistant strain in the steady state. This explains whythe wild-type virus dominates the virus population before the initiation of ART.

Figure 1 presents some simulation results for the CD4+ T cells count and viralloads of both strains before treatment (with the parameters from Table 1). The initial


Fig. 1 Simulation of uninfected T cells and viral loads of both wild-type and drug-resistant strains forthe pretreatment model (2) with initial values: T (0) = 106 ml−1, Vs(0) = 10−6 ml−1, Ts(0) = Tr (0) =Vr (0) = 0. The basic reproductive ratios for the two strains are Rs = 3.13 and Rr = 1.74. Steadystates are T = 3.19 × 105 ml−1, Ts = 6.81 × 103 ml−1, Vs = 8.88 × 105 ml−1, Tr = 0.46 ml−1,Vr = 39.95 ml−1. Before treatment, both strains of virus coexist. However, the drug-resistant strain re-mains at a very low level after an initial increase and the wild-type virus dominates the population.

value of uninfected CD4+ T cells is chosen to be T (0) = 106 ml−1 (Bofill et al., 1992;Perelson et al., 1993; Sedaghat and Siliciano, 2004). The initial viral load Vs(0) is ar-bitrarily set to 10−6 ml−1 (Stafford et al., 2000), which represents a small quantity ofvirions leading to primary infection. The other initial values are set to 0. We observethat the steady state of the resistant viral load is much smaller than that of the wild-typestrain in the absence of therapy. These steady state values of the pretreatment model willbe used as the initial conditions to perform simulations of virus dynamics during drugtherapy.

2.3.2. Stability resultsWe now study the stability of the steady states. We begin with the case of u > 0. Recallthat the coexistence steady state exists if and only if Rs > 1/(1 − u) and σ > 1/(1 − u),and σ > 1/(1 − u), and that the drug-resistant steady state Er exists if and only ifRr > 1.

L. Rong et al.

Linearizing the system (2) about the steady state E, we get

d

dtT (t) = −(d + ksVs + kr Vr )T − ksT Vs − kr T Vr ,

d

dtTs(t) = (1 − u)ksVsT + (1 − u)ksT Vs − δTs,

d

dtVs(t) = NsδTs − cVs,

d

dtTr(t) = (uksVs + kr Vr )T + uksT Vs − δTr + kr T Vr ,

d

dtVr(t) = NrδTr − cVr .

(8)

The corresponding characteristic equation is

∣∣∣∣∣∣∣∣∣

−(d + ksVs + kr Vr ) − ζ 0 −ksT 0 −kr T

(1 − u)ksVs −δ − ζ (1 − u)ksT 0 00 Nsδ −c − ζ 0 0

uksVs + kr Vr 0 uksT −δ − ζ kr T

0 0 0 Nrδ −c − ζ

∣∣∣∣∣∣∣∣∣= 0, (9)

where ζ denotes an eigenvalue of the Jacobian matrix.By considering the characteristic equation (9) at each steady state, we can prove the

following stability results.

Proposition 1. (1) The infection-free steady state E0 is locally asymptotically stable(l.a.s.) if Rs < 1/(1 − u) and Rr < 1, and it is unstable if Rs > 1/(1 − u) or Rr > 1.

(2) The steady state with only drug-resistant virus, Er , exists if and only if Rr > 1. Itis l.a.s. if Rr > (1 − u)Rs and unstable if Rr < (1 − u)Rs .

(3) The coexistence steady state Ec exists and is l.a.s. if and only if Rs > 1/(1 − u)

and Rr < (1 − u)Rs .

Similarly, we have the corresponding stability properties of the steady states for thecase u = 0.

Proposition 2. Let u = 0 in model (2). Then(1) The infection-free steady state E0 is l.a.s if Rs < 1 and Rr < 1, and it is unstable

if Rs > 1 or Rr > 1.(2) The steady state with only drug-resistant virus, Er , exists if and only if Rr > 1. It

is l.a.s. if Rr > Rs and unstable if Rr < Rs .(3) The steady state with only wild-type virus, Es , exists if and only if Rs > 1. It is

l.a.s. if Rs > Rr and unstable if Rs < Rr .

All the stability results for the pretreatment model (2) are summarized in Fig. 2.


Fig. 2 Steady states of model (2) and stability regions (the steady state in bold type is stable in thatregion).

2.4. Uncertainty and sensitivity analysis

The analysis in last subsection suggests that the basic reproductive ratio plays an impor-tant role in predicting viral persistence or eradication. An uncertainty analysis is carriedout in this section to assess the variability of the basic reproductive ratio Rs due to theuncertainty in estimating the input model parameters. A sensitivity analysis that extendsthe uncertainty analysis can identify which parameters are important in contributing tothe variability of Rs (Blower and Dowlatabadi, 1994).

For the uncertainty analysis of Rs , we have to assign specific distributions to para-meters ks , Ns , λ, d and c. Here ks is sampled from a uniform distribution over the in-terval (1.2 × 10−8,3.6 × 10−8) ml day−1 to account for 1–3% of its diffusion-limitedvalue (Perelson et al., 1993) and Ns is sampled from another uniform distribution overthe interval (2000,4000) with the mean 3000 that has been assumed in previous simula-tions. We assume an asymmetric triangular distribution (Blower and Dowlatabadi, 1994)(0.005,0.01,0.016) day−1 for d , the death rate of uninfected T cells, where 0.005 is thelower value, 0.01 is the peak value and 0.016 is the upper value (Mohri et al., 1998). λ issampled from a T0 multiple of the above triangular distribution and c is sampled from an-other asymmetric triangular distribution (9.1,23,36) day−1 with the peak value 23 (Ram-ratnam et al., 1999).

We generate 5 Monte Carlo samples of 105 repetitions from the parameter distributionschosen above and use these input vectors to assess the variability of Rs . Statistical resultsare listed in Table 2. The mean and standard deviation of the distribution of Rs are 3.57and 2.09, respectively. The probability of Rs being greater than 1 is 0.98.

To study the sensitivity of Rs due to the uncertainty of the input parameters we as-sess the partial rank correlation coefficients (PRCCs) between Rs and each parameter(Conover, 1980). These coefficients measure the independent influence of each input pa-rameter on the variability of Rs (Blower and Dowlatabadi, 1994). A value close to 1suggests a strong correlation between Rs and the corresponding variable. We list thesecoefficients according to their absolute values in decreasing order in Table 3. We ob-serve that there exists a strong correlation between Rs and each input parameter, and

L. Rong et al.

Table 2 Statistics for Rs from 5 Monte Carlo samples of 105 repetitions

Sample Mean Std. Dev. Median P(Rs > 1)

1 3.571 2.080 3.082 0.9802 3.576 2.101 3.076 0.9803 3.573 2.089 3.086 0.9794 3.573 2.100 3.078 0.9805 3.565 2.095 3.078 0.980

Mean 3.57 2.09 3.08 0.98

Table 3 PRCC value for each parameter

Parameter PRCC p-value

ks 0.914 <0.0001c −0.877 <0.0001λ 0.849 <0.0001d −0.848 <0.0001Ns 0.818 <0.0001

that ks is the most influential variable. Therefore, decreasing the effective infection rate(for example, increasing the drug efficacy of the RT inhibitor, see model (11) in nextsection) is more effective in reducing Rs for the set of parameter values. However, weshould note that the difference of each coefficient for the corresponding parameter is notsignificant and these PRCCs depend on the parameter distributions chosen above. If webroaden the ranges of ks and Ns , e.g., let ks be sampled from a uniform distribution over(1.2 × 10−8,1.2 × 10−7) ml day−1 to account for 1–10% of its diffusion-limited valueand Ns vary uniformly from 100 to 10000, then the mean and standard deviation of thedistributions of Rs are 16.54 and 16.02, respectively. The probability of Rs being greaterthan 1 is 0.96. The PRCC values between Rs and each parameter are listed in decreasingorder: 0.911(Ns), 0.859(ks), −0.593(c), 0.546(λ), −0.545(d). In this case the burst sizeNs is the most influential variable.

3. Model with antiretroviral therapy

In last section, we have shown that for chronically infected HIV-1 patients the drug-resistant strain exists before the initiation of antiretroviral therapy. However, the virallevel of resistant mutants is expected to be very low and wild-type virus dominates thepopulation. Given a small mutation rate u, the steady state of mutant virus can be approx-imated as:

Vr = d(Rs − 1)

kr(σ − 1)u. (10)

The viral load of the resistant strain is proportional to the mutation rate. This implies thatdrug-resistant mutants are less likely to arise if they differ from wild type by two or morepoints mutations. In this section, we analytically study the mechanisms underlying theemergence of resistant strains during ART.


Two classes of ARV drugs are generally used in HIV treatment. One class is reversetranscriptase inhibitors (RTIs), which can effectively block the infection of target cells byfree virus; the other is protease inhibitors (PIs), which prevent HIV protease from cleavingthe HIV polyprotein into functional units, causing infected cells to produce immaturevirus particles that are noninfectious. Let εs

RT, εrRT be the efficacies of RTIs and εs

PI, εrPI be

the efficacies of PIs for the drug sensitive strain and drug-resistant strains, respectively.We incorporate the effect of ARV regimens in the two-strain model (2) and obtain thefollowing equations:

d

dtT (t) = λ − dT − ks

(1 − εs

RT

)VsT − kr

(1 − εr

RT

)VrT ,

d

dtTs(t) = (1 − u)ks

(1 − εs

RT

)VsT − δTs,

d

dtVs(t) = Ns

(1 − εs

PI

)δTs − cVs,

d

dtTr(t) = uks

(1 − εs

RT

)VsT + kr

(1 − εr

RT

)VrT − δTr ,

d

dtVr(t) = Nr

(1 − εr

PI

)δTr − cVr .

(11)

In the above equations, Vs and Vr are used to represent infectious wild-type virus andinfectious drug-resistant virus, respectively. We have left out two equations that representthe noninfectious virus of both strains since they can be decoupled from system (11).

Provided that all the efficacies are constant, the incorporation of these drugs will notaffect our analysis of the steady states of the post-treatment model (11). The new repro-ductive ratios in the presence of drug therapy can be expressed as:

R′s = (

1 − εsRT

)(1 − εs

PI

)Rs , R′

r = (1 − εr

RT

)(1 − εr

PI

)Rr , (12)

where Rs and Rr are given in (6). Similarly, we can define the ratio σ ′ as R′s/R′

r .Because of the reduced viral fitness of the drug-resistant strain compared with the wild-

type strain, Rs > Rr > 1 and hence the wild-type strain dominates the virus populationbefore the onset of therapy. After the initiation of treatment, the reproductive ratios of bothwild-type and drug-resistant strains decrease. However, we have εs

RT > εrRT and εs

PI > εrPI

as the wild type is more susceptible to drugs, thus R′s decreases more than R′

r . For theease of illustration, we define an overall treatment effect for each strain, i.e.,

εs = 1 − (1 − εs

RT

)(1 − εs

PI

), εr = 1 − (

1 − εrRT

)(1 − εr

PI

). (13)

We begin with the assumption εr = αεs , where α (0 < α < 1) represents the resistancelevel of the HIV mutants, a smaller α indicates that the resistant strain is more resistant tothe drug used. Estimates of drug efficacies for the wild-type and the drug-resistant strainswill be provided in the next section. Here we aim to obtain some explicit conditions forthe emergence of drug resistance by assuming a fixed drug efficacy.

Typical scenarios of the variation of reproductive numbers R′s and R′

r are depicted inFig. 3. We observe that both R′

s and R′r decrease as the drug efficacy εs increases. When

L. Rong et al.

Fig. 3 The upper panel: the reproductive ratio of each strain as a function of the overall drug efficacyfor the sensitive strain εs (see (12) and (13)). (a) α = 0.5; (b) α = 0.2. εr is reduced by a factor α,i.e., εr = αεs . For the ease of illustration, we intentionally enlarge the difference between two lines R′

sand (1 − u)R′

s . The lower panel: steady states of the wild-type and resistant virus as the function ofreproductive ratios, R′

s and R′r , respectively, where α = 0.5. We observe that the wild-type virus can be

completely suppressed even when the reproductive ratio R′s is greater than 1. The resistant virus dies out

only when R′r < 1, and remains at a very low level (not clearly shown in the figure due to the magnitude

of the vertical axis) when R′r > 1.21 (i.e. when εs < ε1, see Fig. 4(a)).

α is 0.5 (Fig. 3(a)), there are two threshold values for εs . One is ε1, the intersection of(1 − u)R′

s and R′r , which is given by

ε1 = (1 − u)Rs −Rr

(1 − u)Rs − αRr

, (14)

and the other one is ε2 with the expression

ε2 = Rr − 1

αRr

. (15)

If the drug efficacy εs is less than ε1 then (1 − u)R′s > R′

r > 1. Thus, both the wild-type and the drug-resistant strains will coexist (similar to the pretreatment case), and thetreatment fails primarily due to the wild-type virus. It is interesting to explore the variation


Fig. 4 α = 0.5. Steady state of virus of model (11) as a function of the overall drug efficacy for thesensitive strain εs . For simplicity, we choose εs

PI = εrPI = 0. Thus, εs

RT = εs , εrRT = εr . We also assume

that εr = αεs . (a) The steady state of resistant virus as the drug efficacy approaches 0.61 (the thresholdvalue for the two strains to coexist, i.e., σ ′ = R′

s/R′r > 1/(1 −u)). At first, the steady state increases very

slowly (sometimes it decreases) as the therapy becomes more effective. When εs → 0.61, the steady stateincreases substantially to the maximum value. (b) When εs increases from 0.61 to 0.85, the steady state ofresistant virus decreases from its maximum to zero. When the drug efficacy is greater than 0.85, both of thetwo strains die out. (c) Steady state of the wild-type virus as a function of the drug efficacy εs . When thedrug efficacy increases, the wild-type viral level decreases. When approaching the threshold value 0.61,the wild-type virus decreases to 0. (d) Total virus population as the function of drug efficacy. Although thedrug-resistant strains undergo a rapid increase during therapy, the total virus keeps decreasing as the drugefficacy increases. When εs > 0.85, both strains of virus will be eradicated.

of the viral steady state levels for the two strains. When εs is small, the viral load ofthe resistant strain is very low (see (10)) compared with the wild-type virus althoughtwo strains coexist. As the drug efficacy is increased from 0, the steady state level ofthe resistant strain increases very slowly (see Fig. 4(a)), whereas the wild type decreasesquickly (Fig. 4(c)). As εs approaches ε1, σ ′ = R′

s/R′r increases and approaches 1/(1−u).

From the formula for Vr given in (5), a direct calculation shows that the steady state ofthe resistant virus in the presence of therapy increases substantially to (R′

r −1)d/(krεrRT).

Coincidentally, this maximum value of the resistant virus load is identical to the steadystate of the drug-resistant virus when only the mutant strain exists (see (4)). The steadystate of the wild-type virus decreases rapidly to 0 as εs approaches ε1 (see Fig. 4(c)).

L. Rong et al.

Fig. 5 Similar to Fig. 4 except α = 0.2. When the drug efficacy approaches the threshold value 0.49,the sensitive strain goes extinct and the resistant strain undergoes a great increase. Even when the drug is100% effective against the wild-type virus, the (drug-resistant) virus still persists.

Mathematical analysis shows that the slope of the decreasing curve is about the orderof 1/u.

For intermediate values of εs in the interval (ε1, ε2), R′r > 1 and R′

r > (1−u)R′s . Thus,

from Proposition 1 only the drug-resistant virus will persist (see Figs. 2(b) and 4(b)).When the drug efficacy increases from ε1 to ε2, the steady state level of the drug-resistantvirus decreases from its maximum value to 0 (Fig. 4(b)).

It is important to note that under drug therapy when εs > ε1, the wild-type virus can besuppressed even when the reproductive ratio of the wild-type strain is greater than 1 (seeFigs. 3(a, c) and 4(c)). This is not surprising because the two viral strains compete for theexact same resources—uninfected target T cells, hence the resistant strain that becomesmore fit as εs > ε1 will outcompete the sensitive one due to the competitive exclusionprinciple. In Figure 3(c), we plot the steady state of the wild-type virus, Vs , as a functionof the reproductive ratio R′

s . We observe that, as the drug efficacy increases, the steadystate of Vs decreases to 0 even when R′

s is greater than 1. Figure 3(d) is for the steadystate of the resistant virus. The resistant virus dies out only if R′

r < 1. When R′r > 1.21,

the steady state of Vr remains at a very low level, which is not clearly shown in the figuredue to the magnitude of the vertical axis.


Fig. 6 Time evolution of uninfected T cells and both strains of virus of model (11). α = 0.2, εsRT = 0.40,

εrRT = αεs

RT, εsPI = εr

PI = 0. The other parameters can be found in Table 1. The initial condition is the

steady state of the pretreatment model (2), i.e., T = 3.19 × 105 ml−1, Ts = 6.81 × 103 ml−1, Vs =8.88 × 105 ml−1, Tr = 0.46 ml−1, Vr = 39.95 ml−1. Both strains of virus persist and uninfected T cellsconverge to nearly 530 cells/µl.

Finally, for a large value of drug efficacy, εs > ε2, we have that R′r < 1 and R′

s <

1/(1−u). It follows from Proposition 1 that both strains will be eradicated by the therapy(Fig. 4(b, c, d)).

In Fig. 3(b), we use a smaller α (α = 0.2) to illustrate another case in which the mutantstrain is more resistant to the drug than that in Fig. 3(a). We observe that the new repro-ductive ratio of the resistant strain will never decrease to below unity even when the drugefficacy εs increases to 1. Consequently, there is only one threshold value ε1, which isabout 0.49. When εs < 0.49, the two strains coexist (Fig. 5(b, c)), and the wild-type virusdominates the population if εs is not close to 0.49. The steady state level of the resistantvirus undergoes a substantial increase when εs approaches 0.49. When εs > 0.49, onlythe resistant strain can persist (Fig. 5(b, c)).

From Figs. 4(d) and 5(d), we also observe that the steady state of the total virus de-creases as the drug efficacy increases. There is a sharp decrease when εs → ε1. This isdue to the fact that the increase in the drug-resistant virus is less than the decrease in thewild-type virus as εs → ε1. The comparison can be clearly observed from Fig. 5(b, c).

L. Rong et al.

Fig. 7 Similar to Fig. 6 except εsRT = 0.51. Only the resistant strain persists. The evolution of each strain

can be seen clearly. Uninfected T cells oscillate to the steady state: 650 cells/µl.

Fig. 8 Dynamics of uninfected T cells and the drug-resistant virus for different drug efficacies: εsRT = 0.6

and εsRT = 0.8. The other parameters and the initial condition are the same as those in Fig. 6. Only the

resistant strain persists. The larger the drug efficacy, the lower steady state of virus and the higher steadystate of uninfected cells.


The treatment success shown in Fig. 4(d) is because we have assumed that the resistantstrain is still very susceptible to the drug regimen. When it is less susceptible to the drug,the virus can not be eradicated even with a treatment highly effective against the sensitivestrain (see Fig. 5(d)).

Figures 6, 7 and 8 illustrate the dynamics of uninfected T cells and viral loads fordifferent drug efficacies. We use the steady state values of variables in the pretreatmentmodel (Fig. 1) as the initial values in the simulations to study the effect of drug therapyon the evolution of the resistant strains. In all the simulations, α is assumed to be 0.2(see Fig. 3(b)). In Fig. 6, εs = 0.40, both strains of virus coexist, but the wild-type virusdominates the population. Drug-resistant mutants arise several weeks after the initiationof drug treatment, and oscillate to a low level set-point value after more than one year.Figure 7 is for a more effective drug regimen, εs = 0.51. We clearly observe the suppres-sion of the wild-type virus and an increase of the drug resistant virus in Fig. 7(b, c). Theresistant virus emerges gradually and eventually out-competes the wild-type virus. Forhigher values of drug efficacy εs , illustrated in Fig. 8, the wild-type virus is suppressedquickly and only the drug-resistant strain persists. When the drug efficacy increases, wealso observe that the number of uninfected T cells converges to a higher level (Fig. 8(a)),while the amplitude of the (resistant) viral peak and the value at the steady state are de-creased. However, with a higher drug efficacy it takes a longer time for the viral load toreach the first viral peak than that with a lower drug efficacy (Fig. 8(b)).

4. Time-varying drug efficacy and effect of adherence

In our analysis so far, the drug efficacies of RTIs and PIs were assumed to be con-stant for both the wild-type and the drug-resistant strains. This assumption may notbe realistic since drug concentrations continuously vary due to drug absorption, dis-tribution, and metabolism in the body. Another important factor affecting the drug ef-ficacy is nonadherence to the regimen protocol. In clinical practice, it is widely be-lieved that the level of compliance with ARV regimens is one of the crucial de-terminants of a successful treatment (Deeks, 2003; Friedland and Williams, 1999;Mugavero and Hicks, 2004). Suboptimal adherence is associated with the emergence ofdrug resistance, viral rebound, and consequently an increased risk of transmitting drug-resistant virus (Bajaria et al., 2004; Deeks et al., 2003; Tesoriero et al., 2003). In thissection, we employ a pharmacokinetic model to determine drug efficacies for both strainsof virus. Then different patterns of non-adherence to drug regimens are considered in or-der to study the effects of time-varying drug efficacies on the emergence of drug-resistantvirus.

4.1. Models for drug efficacy and adherence

There are some existing models that use the plasma drug concentrations to deter-mine the efficacy of antiviral treatment (Dixit et al., 2004; Dixit and Perelson, 2004;Huang et al., 2003; Wahl and Nowak, 2000; Wu et al., 2005). Huang et al. used a stan-dard pharmacokinetic one-compartment model to estimate the drug efficacy and studiedhow drug pharmacokinetics affects antiviral response (Huang et al., 2003). However, it isthe intracellular concentration rather than plasma concentration of drugs that determines

L. Rong et al.

drug effectiveness. Thus we will employ a two-compartment pharmacokinetic model de-veloped recently by Dixit and Perelson (2004) to estimate the treatment efficacy of twoARV drugs, tenofovir disoproxil fumarate (an RTI) and ritonavir (a PI), for wild-type anddrug-resistant strains.

We first describe the two-compartment model briefly (refer to Dixit and Perelson, 2004for a detailed description of the model formulation).

The instantaneous drug efficacy ε(t) can be estimated by a simple function (Dixit et al.,2004; Gabrielson and Weiner, 2000)

ε(t) = Cc(t)

IC50 + Cc(t), (16)

where Cc(t) is the intracellular concentration of the drug used, IC50 is a phenotype markerrepresenting the intracellular concentration of drug needed to inhibit viral replicationby 50%.

If multiple doses of a drug are administered on a regular schedule, then the concentra-tion of drug in the blood is given by

Cb(t) = FDkae−ket

Vd(ke − ka)(ekaId − 1)

[1 − e(ke−ka)t (1 − eNdkaId )

+ (ekeId − ekaId )(e(Nd−1)keId − 1)

ekeId − 1− e((Nd−1)ke+ka)Id

], (17)

where F is the bioavailability of drug, D is the mass of drug administered in one dose,Vd is the volume of distribution, ka and ke are pharmacokinetic parameters that can beestimated from experiments. Id is the dosing interval and Nd is the number of doses untiltime t .

For PIs, the intracellular concentration, Cc , can be derived directly according to thedrug transport from the blood into the cell compartment:

dCc

dt= ka cellCx − ke cellCc, (18)

where

Cx ={

(1 − fb)HCb − Cc if (1 − fb)HCb − Cc > 0,

0 else.(19)

In the above equations, drug absorption is being driven by an effective concentrationgradient. H quantifies the drug partitioning effect of the cell membrane, fb denotes thefraction of drug that can not be transported into cells due to binding plasma proteins, ka cell

and ke cell represent the cellular absorption and elimination rate constants, respectively. See(Dixit and Perelson, 2004) for further details.

For RTIs, the drug action is more complicated. They need to be phosphorylated to theiractive forms in cells. Dixit and Perelson (2004) used the following equations to model thephosphorylation of tenofovir disoproxil fumarate (DF):


dCc

dt= ka cellCx − ke cellCc − k1f Cc + k1bCcp,

dCcp

dt= −ke cellCcp + k1f Cc − k1bCcp − k2f Ccp + k2bCcpp, (20)

dCcpp

dt= −ke cellCcpp + k2f Ccp − k2bCcpp,

where Cc , Ccp and Ccpp represent the respective intracellular concentrations of the native(monophosphorylated), diphosphorylated and triphosphorylated forms of the drug, Cx isgiven in (19), k1f , k1b , k2f and k2b characterize the phosphorylation reactions among Cc ,Ccp and Ccpp .

Solving (20) with the initial condition

Cc(0) = Ccp(0) = Ccpp(0) = 0

and substituting Ccpp for Cc in (16), we obtain the time-dependent efficacy of the RT in-hibitor, which is plotted in Fig. 9(a). Here we choose parameters characteristic of tenofovirDF (Dixit and Perelson, 2004): D = 300 mg, Id = 1 day, F = 0.39, Vd = 87500 ml, ka =14.64 day−1, ke = 9.6 day−1, H = 1800, fb = 0.07, k1f = 9.6 day−1, k1b = 30.3 day−1,k2f = 270.7 day−1, k2b = 95.5 day−1, ka cell = 24000 day−1, ke cell = 1.1 day−1, IC50 is0.54 mg ml−1 for the wild-type strain.

Similarly, solving (18) with initial condition Cc(0) = 0 and plugging the solutioninto (16), we get the efficacy of the protease inhibitor, which is plotted in Fig. 9(b).The parameters characteristic of ritonavir are (Dixit and Perelson, 2004): D = 600 mg,Id = 0.5 days, F = 1, Vd = 28000 ml, ka = 14.64 day−1, ke = 6.86 day−1, H = 0.052,fb = 0.99, IC50 is 9 × 10−7 mg ml−1 for the wild-type strain.

In Fig. 9, we also estimate the drug efficacy for the resistant strain with perfect or sub-optimal drug adherence. We assume that drug resistance increases the virus IC50 10-fold(Larder and Kemp, 1989). The efficacy for the resistant strain is plotted in Fig. 9(a, b) forcomparison with that of the wild-type strain. Figure 9(c, d) plots the drug efficacy whenevery other dose of tenofovir DF and ritonavir is missed. It is clear that when a dose ofdrug is missed, the drug efficacy decreases to a very low level before the next dose isadministered.

As a comparison, we present in Fig. 9(e, f) the time-dependent efficacies of tenofovirDF and ritonavir using the plasma drug concentration rather than the intracellular con-centration in (16). For tenofovir DF, the plasma drug concentration decays more rapidlythan the intracellular concentration. The IC50 for the wild-type strain corresponding tothe plasma concentration is 3.6 × 10−4 mg ml−1 (Dixit and Perelson, 2004). The efficacyεRT is depicted in Fig. 9(e). Using the area under the efficacy curve, we can also estimatethe average drug efficacy. From Fig. 9(a, e) we observe that the average drug efficacy us-ing the plasma concentration is less than that using the intracellular concentration. On thecontrary, the drug concentration of ritonavir in plasma remains sufficiently high comparedwith the corresponding IC50 value, 1.7 × 10−5 mg ml−1. Thus the efficacy εPI using theplasma drug concentration is ∼1 (Dixit and Perelson, 2005), which is plotted in Fig. 9(f).

L. Rong et al.

Fig. 9 The top and middle panels: drug efficacy using the intracellular concentration; the bottom panel:drug efficacy using the plasma concentration. The left column: drug efficacy εRT; the right column: drugefficacy εPI. The horizontal line represents the average drug efficacy. See the text for parameter values.(a, b, e, f) Standard daily dosing. (c) Every other dose of tenofovir DF is missed, i.e., Id = 2 days. (d) Everyother dose of ritonavir is missed, i.e., Id = 1 day.

4.2. Effects on virus dynamics

We conduct numerical simulations to study the time evolution of uninfected T cells andtwo strains of virus based on the time-varying drug efficacies obtained in the previoussubsection.


Fig. 10 Dynamics with perfect adherence. The solid line represents the dynamics of uninfected T cells andvirus when drug resistance increases the baseline IC50 10-fold. The dashed line is for 15-fold resistance.In both cases, wild-type virus is predicted to be eradicated due to the perfect adherence to ART (in realityviral eradication is not observed due to viral reservoirs, such as latently infected cells, which are notdiscussed in the model). The higher resistance level of the mutant strain, the more quickly drug-resistantvirus will emerge and dominate. The viral peak and the final periodic orbit level of drug-resistant virus arealso increased when the resistance level increases. All the variables converge to periodic orbits with veryfrequent oscillations because of the time-dependent drug efficacy.

We first consider the scenario of perfect adherence to the prescribed dose levels anddosing times. We consider combination therapy with both the RTI and PI drugs and as-sume that the mutant strain is resistant to both of them. As point mutations can conferdifferent levels of resistance to drug (Larder, 1996), we study two cases in which drugresistance increases the baseline IC50 by different amounts. In Case 1, drug resistance isassumed to increase the IC50 value 10-fold for both the RTI and PI; Case 2 deals with a15-fold increase of IC50 for the resistant strain. Figure 10(a) presents the time evolutionof uninfected T cells during therapy. We observe that the number of uninfected T cells inboth cases oscillates to a periodic orbit lower than the normal CD4+ T level in uninfectedindividuals (1000 cells/µl). Case 2 (15-fold resistance) has a lower periodic orbit levelof uninfected T cells than that of Case 1 (10-fold resistance). Figure 10(b) shows that theviral load of the wild-type strain can be suppressed very well for each case if HIV-infectedindividuals adhere to the treatment perfectly. The drug-resistant virus appears about oneyear after the initiation of therapy in Case 1, while the emergence of resistance in Case 2

L. Rong et al.

is much earlier (see Fig. 10(c)). We can also observe that if the mutant virus is more resis-tant to drugs, then the viral peak and the final periodic orbit level are higher than that ofthe strain with a lower resistance level. In fact, if drug resistance increases the IC50 valueonly 5-fold or less, then both strains will be suppressed if perfect adherence is followed(figure not shown). The viral levels in Fig. 10 include noninfectious virus since a PI ispresent in the treatment.

When we employ the drug efficacy estimated from the plasma concentration(Fig. 9(e, f)), both the (infectious) wild-type and resistant virus will be completely sup-pressed because of the nearly perfect drug effectiveness of the protease inhibitor (figurenot shown). Here we assumed that drug resistance increases the IC50 value 10-fold. Evenif the mutant strain is highly resistant to the protease drug (for example, 100-fold re-sistance), both strains are still predicted to die out (figure not shown). This observationsuggests that the two-compartment pharmacokinetic model developed in (Dixit and Perel-son, 2004) might be more appropriate to estimate the drug efficacy of ritonavir. Althoughit is possible to obtain a good approximation of the intracellular efficacy by use of thefunction of the plasma concentration like (16) and its variations (Huang et al., 2003),the approximation does not capture the pharmacokinetic characteristics, e.g., drug trans-port between the blood and the cell compartment, sequential phosphorylation reactions ofRTIs within cells.

In Fig. 11, we simulate the viral response under imperfect adherence. We still considertwo cases. In the first case, the individual misses every other dose of the RT drug andprotease drug. In the second case, the individual misses more doses: e.g., one dose of RTinhibitor is taken and followed by two missed doses; and one dose of protease inhibitoris taken and followed by three missed doses. In both cases, drug resistance is assumed toincrease the baseline IC50 10-fold. The dynamics of uninfected T cells are illustrated inFig. 11(a). When more doses are missed, the number of uninfected T cells converges toa lower periodic orbit level. Figure 11(b) illustrates that the wild-type virus can still besuppressed well if every other dose of both drugs is missed. However, if more doses aremissed, the wild-type virus can not be eradicated. Figure 11(c) shows the emergence ofthe drug-resistant virus during treatment. When every other dose is missed, resistant virusemerges quickly and substantially after the therapy compared with the case of perfectadherence (Fig. 10(c)). If more doses are missed, drug resistant virus arises very slowlyand the viral load is kept at a low level for several years. The wild-type virus out-competesthe resistant virus. This is similar to the case discussed in previous sections, in which drugtreatment is less effective and the viral load of the resistant strain remains at a low levelalthough two strains coexist. Due to the periodic drug dosing schedule all the curvesundergo frequent oscillations converging to periodic orbits. When more doses are missed,the oscillation of the total viral load can be seen clearly from Fig. 11(d).

4.3. Adherence pattern and average drug efficacy

The proceeding result shows that the emergence of the drug resistant virus depends heav-ily on the relative susceptibilities of the two strains to drug therapy and the level of patientadherence. If we use p, the fraction of prescribed doses taken, to characterize the degreeof adherence to a specific drug (Wahl and Nowak, 2000), then in last simulation (Fig. 11)p = 1/2 when every other dose is missed, p = 1/3 when one dose is taken and followedby two missed doses. Simulation results suggest that the resistant virus is more likely to


Fig. 11 Dynamics with suboptimal adherence. Drug resistance is assumed to increase the virus IC5010-fold. The solid line represents the dynamics of uninfected T cells and virus when every other dose ofboth drugs is missed. Wild-type virus is eradicated and drug resistance arises quickly and substantially.The dashed line illustrates the situation when more doses are missed (see the text for description). Thetreatment can not suppress the wild-type strain and the drug-resistant virus increases very slowly andremains at a low level.

emerge and stay at a relatively high level for intermediate values of p. When p is verylarge, the resistant virus is predicted to be eradicated; when p is very small, the resistantvirus remains at a very low level. These results are consistent with the observations in(Wahl and Nowak, 2000).

Now a natural question arises: given the same degree of adherence, does the adherencepattern affect virus dynamics and long-term prediction of therapy? We attempt to addressthis question in the following.

For simplicity, we just consider monotherapy with the RT inhibitor, tenofovir DF. InFig. 12, we present four adherence patterns with the same value of p (p = 1/2). Fig-ure 12(a) shows the drug efficacies of both strains when every other dose is missed(Pattern 1). Pattern 2 shows two doses are taken followed by two missed doses. Pat-tern 3 shows three doses are taken followed by three missed doses and Pattern 4 showsfive doses are taken followed by five missed doses. In fact, Patterns 1–4 describe dif-ferent adherence patterns by the block size, which is defined as the number of consec-utive doses taken or missed each time a dose is taken or missed (Huang et al., 2003;

L. Rong et al.

Fig. 12 Drug efficacy of tenofovir DF for different adherence patterns with the same degree of adherencep = 1/2. (a) Every other dose is missed. (b) Two doses are taken followed by two missed doses. (c) Threedoses are taken followed by three missed doses. (d) Five doses are taken followed by five missed doses.The average drug efficacy is decreasing as the block size increases.

Wahl and Nowak, 2000). The block size for Patterns 1, 2, 3, 4 are 1, 2, 3, 5, respectively.It is interesting to note that, as shown in Fig. 12, the average drug efficacy decreasesslightly when the block size increases.

As an illustration, we present in Fig. 13 the time evolution of the viral load and unin-fected T cells for Pattern 1 and Pattern 4. From Fig. 13(a) we observe clearly that Pattern 1performs better than Pattern 4 in keeping uninfected T cells at a higher level. However,the final periodic orbit level of the resistant virus with Pattern 4 is lower than that withPattern 1 (Fig. 13(c)). From Fig. 13(b, d) we cannot tell which pattern is better in reducingthe wild-type virus or total virus due to frequent oscillations. However, the average levelsof the wild-type virus and total virus using Pattern 1 are lower than the viral levels usingPattern 4. Thus Pattern 1 seems to perform better than Pattern 4 in increasing uninfectedT cells and reducing total viral load.

The above comparison can be observed clearly if we use the average drug efficacyrather than the time-dependent efficacy to study the viral load and uninfected T cell level.In fact, using the average drug efficacy can give a good prediction of the long-term out-come of therapy although uninfected T cells and the viral load undergo frequent oscilla-tions when the time-dependent drug efficacy is employed (figure not shown). When theblock size increases, the average drug efficacy decreases. Therefore, the final steady states


Fig. 13 Time evolution of uninfected T cells and the viral load for Patterns 1 and 4 shown in Fig. 12. Herewe just consider monotherapy with the RT inhibitor. Pattern 1 seems to perform better than Pattern 4 inincreasing uninfected T cells and reducing total viral load.

of uninfected T cells and the resistant virus decrease, whereas the steady states of thewild-type virus and total virus increase (see Fig. 4). They are consistent with the aboveobservations. This implies that the analysis using constant drug efficacy in Section 3 isstill helpful in predicting which periodic orbit the system variable will approach when weuse the time-varying drug efficacy.

5. Discussion

It is widely recognized that long-term antiretroviral treatment with incompletely sup-pressive regimens is associated with the emergence of drug-resistant HIV mutants.Resistance is the consequence of the selection of mutations in the genes coding for HIV-1 reverse transcriptase or protease. Although there are models in the literature study-ing the emergence of drug resistance in the course of therapy, many of them concernthe likelihood of mutant variants preexisting before treatment as opposed to them be-ing produced during therapy (Bonhoeffer et al., 1997; Bonhoeffer and Nowak, 1997;Ribeiro and Bonhoeffer, 2000; Ribeiro et al., 1998). In this paper, we employ a two-strain mathematical model to study the mechanistic basis of the emergence of resistant

L. Rong et al.

strains under antiretroviral treatment. The following issues regarding virus dynamics andantiretroviral responses have been investigated: understanding the evolution of both wild-type and mutant strains before and during drug therapy; deriving conditions under whichmutant variants are selected in the presence of drug pressure and dominate the virus popu-lation; incorporating realistic pharmacokinetics and adherence behavior to study antiviralresponse and predict HIV treatment outcome. Analytical results show that notwithstand-ing the coexistence of both wild-type and mutant strains the viral level of mutant strain isvery low compared with wild-type strain before treatment or when the treatment is poorlyeffective. Drug resistance is more likely to arise for intermediate levels of treatment ef-fectiveness, at which the reproductive ratios of both strains are close.

Although the model predicts that virus could be eradicated if the antiretroviral treat-ment is potent enough, there are many viral and host factors that might hamper treatmentsuccess. Even with effective ARV drugs, HIV-1 may replicate in body sites lacking ad-equate drug exposure to ARV drugs for the selection of drug-resistant mutants (Zhanget al., 1999). Kepler and Perelson (1998) showed that there is a relatively narrow win-dow of drug concentrations that favors the evolution of resistant virus if one considers thebody as a single compartment. However, the window of opportunity for the generation ofresistance is significantly widened if spatial heterogeneity is taken into account. In someregions, such as the brain and testes, the drug concentration may be low (drug sanctuary),which enables resistant mutants to be generated more easily. Therefore, the model pre-sented in this paper might underestimate the range of drug concentrations that allow theemergence of drug resistance.

In our model, we have assumed that both the clearance of free virus and the deathrate of productively infected cells are the same for the wild-type and resistant strains.However, they could be different for the two strains. The viral clearance could be theresult of natural death, immune elimination, or binding and entry into cells (Perelsonet al., 1996), which might be different for the two strains. The cell death rate, δ, couldbe an increasing function with respect to the viral production rate (Coombs et al., 2003).However, the main purpose of this paper is to investigate antiretroviral responses duringtreatment. We aim to address the question how the resistant virus arises and is selected inthe presence of sufficient drug pressure. Therefore, the choice of these parameters as wellas the infection rate and the burst size will not affect the qualitative property illustrated inFig. 3 as long as the pretreatment reproductive ratios of two strains satisfy Rs > Rr > 1.

Another simplification of the two-strain model is that we have not considered latentlyinfected cells or long-lived infected cells. When the steady state is perturbed with combi-nation therapy, the initial decrease of viral load is usually followed by a slower second-phase viral decay. A number of sources might contribute to the second phase, for example,long-lived infected cells with a half-life of 1–4 weeks (Perelson et al., 1997), virus parti-cles released from follicular dendritic cells in peripheral lymphoid tissues (Hlavacek et al.,2000; Hlavacek et al., 1999), and the activation of latently infected lymphocytes to pro-ductively infected cells (Perelson et al., 1997). We have not considered latently infectedcells in this paper as they do not seem to play a large role in the second-phase viral decline(Perelson et al., 1997) although the persistence of HIV reservoirs, including latently in-fected resting CD4+ T cells, for a prolonged period of time remains a big challenge to thelong-term control or eradication of HIV-1 in infected patients receiving potent antiretro-viral treatment (see a recent review in Kim and Perelson, 2006). For long-lived infectedcells, numerical investigation in (Wein et al., 1998) shows that they have little effect on


the dynamics of mutant virus when eradication does not occur. Whether the incorporationof these cells as well as immune response will affect the emergence of drug resistance andthe prediction of long-term treatment effect awaits future studies.

The level of adherence to the prescribed antiretroviral regimen is very importantamong the determinants of drug therapy. In this paper we have considered two pat-terns of imperfect adherence. In one pattern, infected individuals miss every other dosefor each drug (if p denotes the fraction of the prescribed doses of the drug that aretaken (Huang et al., 2003; Wahl and Nowak, 2000), then p = 0.5 in this case); inthe other pattern, more doses are missed (p < 0.5). Simulation results show that thewild-type virus and the mutant variants with a low level of resistance (for example,5-fold drug resistance) will be suppressed well even if every other dose is missed. How-ever, the mutant strain with a high level of resistance (10-fold resistance in the sim-ulation) will flourish several months after the initiation of antiretroviral treatment. Ifmore doses are missed, resistant strains evolve slowly while the wild-type virus domi-nates the population (see Fig. 11). We realize that the patterns of adherence discussedin this paper are not very realistic. In fact, quantifying the real-world patterns of adher-ence and their influence on the effect of antiretroviral treatment is far from straightfor-ward (Ferguson et al., 2005). Even for the same fraction of the prescribed doses that aretaken, different adherence patterns (for example, different block sizes) can induce dif-ferent treatment outcomes (Huang et al., 2003; Wahl and Nowak, 2000). Future workis required to combine pharmacokinetics with careful modeling of more realistic adher-ence patterns to better predict the antiretroviral responses, particularly the evolution ofdrug resistance. This might help provide a framework to improve the treatment bene-fits through structured treatment interruptions (STIs) (Deeks et al., 2003; Gulick, 2002;Lori et al., 2000; Ortiz et al., 2001). The observation that drug-resistant virus declinesto a low level after HIV-1-infected patients discontinue antiretroviral treatment for aperiod leads to the hypothesis that STIs could be served as a new treatment protocolto achieve similar clinical benefits while allowing patients drug holidays (Deeks, 2003;Miller et al., 2000). However, more attention needs to be paid when designing treatmentschedule through STIs as some patients undergo virus rebound within days during STIs(Fischer et al., 2003). A review of different responses to STIs during therapy can be foundin (Bajaria et al., 2004).

The present model assumes that the drug-resistant strain and the wild-type strain differby a single point mutation. Sometimes substitutions of single amino acids can confer ahigh level of drug resistance and the strain carrying any single mutation is believed toexist before the onset of therapy (Clavel and Hance, 2004; Coffin, 1995). Investigationsin this paper have also verified the preexistence of this type of mutant strain. On the otherhand, if single mutation only generates a low level of resistance to some agents, then highlevels of resistance or complete resistance requires gradual accumulation of additionalmutations (Clavel and Hance, 2004). In these cases, we still can obtain some informationabout those mutant strains from the current model. We make an approximation that thedrug-resistant strain with multiple mutations is mutated directly from the wild-type strainwith a correspondingly smaller mutation rate. There are two possibilities: one is that themultiple mutations confer an intermediate level of resistance (that is higher than that withsingle mutation), and the other one is that a high level of resistance is generated. In thefirst case, mathematical analysis and numerical simulations show that the viral load of thewild-type strain remains at the same level, while the steady state level of the resistant strain

L. Rong et al.

with multiple mutations is much lower than that with a single mutation due to the smallmutation rate. In the second case, only the resistant strain will persist and the viral loadof HIV variants with multiple mutations is higher than that with a single mutation. Thisis because the resistant strain with multiple mutations is less susceptible to the treatment.All these conclusions can be drawn directly from the analysis of expressions of steadystates given in (4) and (5).

One of the important causes leading to treatment failure in HIV-1-infected individualsis the large diversity of HIV genotypes (Vergu et al., 2002), particularly all the drug-resistant mutants that are selected in the presence of drug pressure (as discussed in thispaper). Another source contributing to the resistance that cannot be ignored is the trans-mission of drug-resistant HIV to susceptible individuals (see the recent review in Tangand Pillay, 2004). HIV resistance testing might be a good way to overcome this problem.Currently, there are two methods to investigate HIV resistance: genotypic assays and phe-notypic assays (Richman, 2000). Genotypic assays detect key resistance-associated mu-tations in the reverse transcriptase and protease genes, while phenotypic assays measurethe susceptibility of the virus to antiretroviral drugs in cell culture. Although the asso-ciation between genotypic and phonotypic resistance still remains unclear (Dunn et al.,2004), and in some cases genotypic testing might lead to wrong decisions when choosingtherapy (Roberts and Ribeiro, 2001), HIV resistance testing could provide worthwhileinformation that potentially could allow the design of efficient individual strategies ofantiretroviral treatment.

Acknowledgements

Portions of this work were performed under the auspices of the US Department of Energyunder contract DE-AC52-06NA25396. This work was supported by NSF grant DMS-0314575 and James S. McDonnell Foundation 21st Century Science Initiative (Z.F.), andNIH grants AI28433 and RR06555 (A.S.P.). The manuscript was finished when L.R. vis-ited the Theoretical Biology and Biophysics Group, Los Alamos National Laboratory in2006. L.R. would like to thank A.S.P. for his hospitality and support. L.R. also thanksXiaohong Wang and Pei Zhang for the discussion on the sensitivity analysis. The authorsalso thank two anonymous referees for their constructive comments that greatly improvedthis paper.

References

Bajaria, S.H., Webb, G.F., Kirschner, D.E., 2004. Predicting differential responses to structured treatmentinterruptions during HAART. Bull. Math. Biol. 66, 1093–1118.

Bangsberg, D.R., Perry, S., Charlebois, E.D., Clark, R.A., Roberston, M., Zolopa, A.R., Moss, A., 2001.Non-adherence to highly active antiretroviral therapy predicts progression to AIDS. AIDS 15, 1181–1183.

Barbour, J.D., Wrin, T., Grant, R.M., Martin, J.N., Segal, M.R., Petropoulos, C.J., Deeks, S.G., 2002.Evolution of phenotypic drug susceptibility and viral replication capacity during long-term virologicfailure of protease inhibitor therapy in human immunodeficiency virus-infected adults. J. Virol. 76,11104–11112.

Blower, S.M., Aschenbach, A.N., Gershengorn, H.B., Kahn, J.O., 2001. Predicting the unpredictable:transmission of drug-resistant HIV. Nat. Med. 7, 1016–1020.


Blower, S.M., Dowlatabadi, H., 1994. Sensitivity and uncertainty analysis of complex models of diseasetransmission: an HIV model, as an example. Int. Stat. Rev. 62, 229–243.

Bofill, M., Janossy, G., Lee, C.A., MacDonald-Burns, D., Phillips, A.N., Sabin, C., Timms, A., Johnson,M.A., Kernoff, P.B., 1992. Laboratory control values for CD4 and CD8 T lymphocytes: implicationsfor HIV-1 diagnosis. Clin. Exp. Immunol. 88, 243–252.

Bonhoeffer, S., May, R.M., Shaw, G.M., Nowak, M.A., 1997. Virus dynamics and drug therapy. Proc. Natl.Acad. Sci. USA 94, 6971–6976.

Bonhoeffer, S., Nowak, M.A., 1997. Pre-existence and emergence of drug resistance in HIV-1 infection.Proc. Roy. Soc. Lond. B 264, 631–637.

Clavel, F., Hance, A.J., 2004. HIV drug resistance. New Engl. J. Med. 350, 1023–1035.Clavel, F., Race, E., Mammano, F., 2000. HIV drug resistance and viral fitness. Adv. Pharmacol. 49, 41–

66.Coffin, J.M., 1995. HIV population dynamics in vivo: implications for genetic variation, pathogenesis, and

therapy. Science 267, 483–489.Collier, A.C., Coombs, R.W., Schoenfeld, D.A., Bassett, R.L., Timpone, J., Baruch, A., Jones, M., Facey,

K., Whitacre, C., McAuliffe, V.J., Friedman, H.M., Merigan, T.C., Reichman, R.C., Hooper, C.,Corey, L., 1996. Treatment of human immunodeficiency virus infection with saquinavir, zidovudine,and zalcitabine. New Engl. J. Med. 334, 1011–1017.

Conover, W.J., 1980. Practical Nonparametric Statistics, 2rd edn. Wiley, New York.Coombs, D., Gilchrist, M.A., Percus, J., Perelson, A.S., 2003. Optimal viral production. Bull. Math. Biol.

65, 1003–1023.Deeks, S.G., 2003. Treatment of antiretroviral-drug-resistant HIV-1 infection. Lancet 362, 2002–2011.Deeks, S.G., Grant, R.M., Wrin, T., Paxinos, E.E., Liegler, T., Hoh, R., Martin, J.N., Petropoulos, C.J.,

2003. Persistence of drug-resistant HIV-1 after a structured treatment interruption and its impact ontreatment response. AIDS 17, 361–370.

De Jong, M.D., Veenstra, J., Stilianakis, N.I., Schuurman, R., Lange, J.M., De Boer, R.J., Boucher, C.A.,1996. Host-parasite dynamics and outgrowth of virus containing a single K70R amino acid change inreverse transcriptase are responsible for the loss of human immunodeficiency virus type 1 RNA loadsuppression by zidovudine. Proc. Natl. Acad. Sci. USA 93, 5501–5506.

De Leenheer, P., Smith, H.L., 2003. Virus dynamics: a global analysis. SIAM J. Appl. Math. 63, 1313–1327.

Dixit, N.M., Markowitz, M., Ho, D.D., Perelson, A.S., 2004. Estimates of intracellular delay and averagedrug efficacy from viral load data of HIV-infected individuals under antiretroviral therapy. Antivir.Ther. 9, 237–246.

Dixit, N.M., Perelson, A.S., 2004. Complex patterns of viral load decay under antiretroviral therapy: in-fluence of pharmacokinetics and intracellular delay. J. Theor. Biol. 226, 95–109.

Dixit, N.M., Perelson, A.S., 2005. Influence of drug pharmacokinetics on HIV pathogenesis and therapy.In: Tan, W.-Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS and HIV with Interven-tion, pp. 287–311. World Scientific, Singapore.

Dunn, D.T., Gibb, D.M., Babiker, A.G., Green, H., Darbyshire, J.H., Weller, I.V., 2004. HIV resistancetesting: is the evidence really there? Antivir. Ther. 9, 641–648.

Ferguson, N.M., Donnelly, C.A., Hooper, J., Ghani, A.C., Fraser, C., Bartley, L.M., Rode, R.A., Vernazza,P., Lapins, D., Mayer, S.L., Anderson, R.M., 2005. Adherence to antiretroviral therapy and its impacton clinical outcome in HIV-infected patients. J. Roy. Soc. Interface 2, 349–363.

Fischer, M., Hafner, R., Schneider, C., Trkola, A., Joos, B., Joller, H., Hirschel, B., Weber, R., Gunthard,H.F., Swiss HIV Cohort Study, 2003. HIV RNA in plasma rebounds within days during structuredtreatment interruptions. AIDS 17, 195–199.

Friedland, G.H., Williams, A., 1999. Attaining higher goals in HIV treatment: the central importance ofadherence. AIDS 13(Suppl. 1), S61–S72.

Gabrielson, J., Weiner, D., 2000. Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts andApplications. Swedish Pharmaceutical Press, Stockholm.

Gulick, R.M., 2002. Structured treatment interruption in patients infected with HIV. Drugs 62, 245–253.Haase, A.T., Henry, K., Zupancic, M., Sedgewick, G., Faust, R.A., Melroe, H., Cavert, W., Gebhard, K.,

Staskus, K., Zhang, Z.Q., Dailey, P.J., Balfour, H.H. Jr., Erice, A., Perelson, A.S., 1996. Quantitativeimage analysis of HIV-1 infection in lymphoid tissue. Science 274, 985–989.

Havlir, D.V., Eastman, S., Gamst, A., Richman, D.D., 1996. Nevirapine-resistant human immunodefi-ciency virus: kinetics of replication and estimated prevalence in untreated patients. J. Virol. 70, 7894–7899.

L. Rong et al.

Heffernan, J.M., Wahl, L.M., 2005. Treatment interruptions and resistance: a review. In: Tan, W.-Y., Wu, H.(Eds.), Deterministic and Stochastic Models of AIDS and HIV with Intervention, pp. 423–456. WorldScientific, Singapore.

Hlavacek, W.S., Stilianakis, N.I., Notermans, D.W., Danner, S.A., Perelson, A.S., 2000. Influence of fol-licular dendritic cells on decay of HIV during antiretroviral therapy. Proc. Natl. Acad. Sci. USA 97,10966–10971.

Hlavacek, W.S., Wofsy, C., Perelson, A.S., 1999. Dissociation of HIV-1 from follicular dendritic cellsduring HAART: mathematical analysis. Proc. Natl. Acad. Sci. USA 96, 14681–14686.

Hockett, R.D., Kilby, J.M., Derdeyn, C.A., Saag, M.S., Sillers, M., Squires, K., Chiz, S., Nowak, M.A.,Shaw, G.M., Bucy, R.P., 1999. Constant mean viral copy number per infected cell in tissues regardlessof high, low, or undetectable plasma HIV RNA. J. Exp. Med. 189, 1545–1554.

Huang, Y., Rosenkranz, S.L., Wu, H., 2003. Modeling HIV dynamics and antiviral response with con-sideration of time-varying drug exposures, adherence and phenotypic sensitivity. Math. Biosci. 184,165–186.

Kepler, T.B., Perelson, A.S., 1998. Drug concentration heterogeneity facilitates the evolution of drug re-sistance. Proc. Natl. Acad. Sci. USA 95, 11514–11519.

Kim, H., Perelson, A.S., 2006. Dynamic characteristics of HIV-1 reservoirs. Curr. Opin. HIV AIDS 1,152–156.

Kirschner, D.E., Webb, G.F., 1997. Understanding drug resistance for monotherapy treatment of HIVinfection. Bull. Math. Biol. 59, 763–786.

Larder, B.A., Darby, G., Richman, D.D., 1989. HIV with reduced sensitivity to zidovudine (AZT) isolatedduring prolonged therapy. Science 243, 1731–1734.

Larder, B.A., Kemp, S.D., 1989. Multiple mutations in HIV-1 reverse transcriptase confer high-level resis-tance to zidovudine (AZT). Science 246, 1155–1158.

Larder, B.A., 1996. Nucleosides and foscarnet-mechanisms. In: Richman, D.D., (Ed.), Antiviral DrugResistance. Wiley, New York.

Lori, F., Maserati, R., Foli, A., Seminari, E., Timpone, J., Lisziewicz, J., 2000. Structured treatment inter-ruptions to control HIV-1 infection. Lancet 355, 287–288.

Mansky, L.M., Temin, H.M., 1995. Lower in vivo mutation rate of human immunodeficiency virus type 1than that predicted from the fidelity of purified reverse transcriptase. J. Virol. 69, 5087–5094.

Markowitz, M., Louie, M., Hurley, A., Sun, E., Di Mascio, M., Perelson, A.S., Ho, D.D., 2003. A novelantiviral intervention results in more accurate assessment of human immunodeficiency virus type 1replication dynamics and T-cell decay in vivo. J. Virol. 77, 5037–5038.

McLean, A.R., Nowak, M.A., 1992. Competition between zidovudine-sensitive and zidovudine-resistantstrains of HIV. AIDS 6, 71–79.

Miller, V., Sabin, C., Hertogs, K., Bloor, S., Martinez-Picado, J., D’Aquila, R., Larder, B., Lutz, T., Gute, P.,Weidmann, E., Rabenau, H., Phillips, A., Staszewski, S., 2000. Virological and immunological effectsof treatment interruptions in HIV-1 infected patients with treatment failure. AIDS 14, 2857–2867.

Mohri, H., Bonhoeffer, S., Monard, S., Perelson, A.S., Ho, D.D., 1998. Rapid turnover of T lymphocytesin SIV-infected rhesus macaques. Science 279, 1223–1227.

Mugavero, M.J., Hicks, C.B., 2004. HIV resistance and the effectiveness of combination antiretroviraltreatment. Drug Discov. Today Ther. Strateg. 1, 529–535.

Murray, J.M., Perelson, A.S., 2005. Human immunodeficiency virus: quasi-species and drug resistance.Multiscale Model. Simul. 3, 300–311.

Nowak, M.A., Bonhoeffer, S., Shaw, G.M., May, R.M., 1997. Anti-viral drug treatment: dynamics ofresistance in free virus and infected cell populations. J. Theor. Biol. 184, 203–217.

Nowak, M.A., May, R.M., 2000. Virus Dynamics: Mathematical Principles of Immunology and Virology.Oxford University Press, London.

Ortiz, G.M., Wellons, M., Brancato, J., Vo, H.T., Zinn, R.L., Clarkson, D.E., Van Loon, K., Bonhoeffer,S., Miralles, G.D., Montefiori, D., Bartlett, J.A., Nixon, D.F., 2001. Structured antiretroviral treatmentinterruptions in chronically HIV-1-infected subjects. Proc. Natl. Acad. Sci. USA 98, 13288–13293.

Perelson, A.S., Essunger, P., Cao, Y., Vesanen, M., Hurley, A., Saksela, K., Markowitz, M., Ho, D.D., 1997.Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387, 188–191.

Perelson, A.S., Essunger, P., Ho, D.D., 1997. Dynamics of HIV-1 and CD4+ lymphocytes in vivo. AIDS11(Suppl. A), S17–S24.

Perelson, A.S., Kirschner, D.E., De Boer, R., 1993. Dynamics of HIV infection of CD4+ T cells. Math.Biosci. 114, 81–125.


Perelson, A.S., Nelson, P.W., 2002. Modeling viral infections. Proc. Symp. Appl. Math. 59, 139–172.Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D., 1996. HIV-1 dynamics in vivo:

virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586.Phillips, A.N., Youle, M., Johnson, M., Loveday, C., 2001. Use of a stochastic model to develop under-

standing of the impact of different patterns of antiretroviral drug use on resistance development. AIDS15, 2211–2220.

Ramratnam, B., Bonhoeffer, S., Binley, J., Hurley, A., Zhang, L., Mittler, J.E., Markowitz, M., Moore,J.P., Perelson, A.S., Ho, D.D., 1999. Rapid production and clearance of HIV-1 and hepatitis C virusassessed by large volume plasma apheresis. Lancet 354, 1782–1785.

Ribeiro, R.M., Bonhoeffer, S., 2000. Production of resistant HIV mutants during antiretroviral therapy.Proc. Natl. Acad. Sci. USA 97, 7681–7686.

Ribeiro, R.M., Bonhoeffer, S., Nowak, M.A., 1998. The frequency of resistant mutant virus before antiviraltherapy. AIDS 12, 461–465.

Richman, D.D., 1992. Selection of zidovudine-resistant variants of human immunodeficiency virus bytherapy. Curr. Top. Microbiol. Immunol. 176, 131–142.

Richman, D.D., 1996. The implications of drug resistance for strategies of combination antiviralchemotherapy. Antivir. Res. 29, 31–33.

Richman, D.D., 2000. Principles of HIV resistance testing and overview of assay performance character-istics. Antivir. Ther. 5, 27–31.

Richman, D.D., Havlir, D., Corbeil, J., Looney, D., Ignacio, C., Spector, S.A., Sullivan, J., Cheeseman, S.,Barringer, K., Pauletti, D., et al., 1994. Nevirapine resistant mutations of human immunodeficiencyvirus type 1 selected during therapy. J. Virol. 68, 1660–1666.

Roberts, D.E., Ribeiro, R.M., 2001. Comparison of different treatment regimens for the emergence of newresistance under therapy. J. Acquir. Immune Defic. Syndr. 27, 331–335.

Sedaghat, A.R., Siliciano, R.F., 2004. Immunodeficiency in HIV-1 infection. In: Wormser, G. (Ed.), AIDSand Other Manifestations of HIV Infection, 4th edn., pp. 265–283. Elsevier, Amsterdam.

Sethi, A.K., Celentano, D.D., Gange, S.J., Moore, R.D., Gallant, J.E., 2003. Association between adher-ence to antiretroviral therapy and human immunodeficiency virus drug resistance. Clin. Infect. Dis.37, 1112–1118.

Shiri, T., Garira, W., Musekwa, S.D., 2005. A two-strain HIV-1 mathematical model to assess the effectsof chemotherapy on disease parameters. Math. Biosci. Eng. 2, 811–832.

Smith, R.J., 2006. Adherence to antiretroviral HIV drugs: how many doses can you miss before resistanceemerges? Proc. Roy. Soc. B 273, 617–624.

Smith, R.J., Wahl, L.M., 2005. Drug resistance in an immunological model of HIV-1 infection with im-pulsive drug effects. Bull. Math. Biol. 67, 783–813.

Snedecor, S.J., 2003. Comparison of three kinetic models of HIV-1 infection: implications for optimizationof treatment. J. Theor. Biol. 221, 519–541.

Stafford, M.A., Corey, L., Cao, Y., Daar, E.S., Ho, D.D., Perelson, A.S., 2000. Modeling plasma virusconcentration during primary HIV infection. J. Theor. Biol. 203, 285–301.

St Clair, M.H., Martin, J.L., Tudor-Williams, G., Bach, M.C., Vavro, C.L., King, D.M., Kellam, P., Kemp,S.D., Larder, B.A., 1991. Resistance to ddI and sensitivity to AZT induced by a mutation in HIV-1reverse transcriptase. Science 253, 1557–1559.

Stilianakis, N.I., Boucher, C.A., De Jong, M.D., Van Leeuwen, R., Schuurman, R., De Boer, R.J., 1997.Clinical data sets of human immunodeficiency virus type 1 reverse transcriptase-resistant mutantsexplained by a mathematical model. J. Virol. 71, 161–168.

Tang, J.W., Pillay, D., 2004. Transmission of HIV-1 drug resistance. J. Clin. Virol. 30, 1–10.Tesoriero, J., French, T., Weiss, L., Waters, M., Finkelstein, R., Agins, B., 2003. Stability of adherence

to highly active antiretroviral therapy over time among clients enrolled in the treatment adherencedemonstration project. J. Acquir. Immune Defic. Syndr. 33, 484–493.

Tchetgen, E., Kaplan, E.H., Friedland, G.H., 2001. Public health consequences of screening patients foradherence to highly active antiretroviral therapy. J. Acquir. Immune Defic. Syndr. 26, 118–129.

Vergu, E., Mallet, A., Golmard, J.L., 2002. The role of resistance characteristics of viral strains in theprediction of the response to antiretroviral therapy in HIV infection. J. Acquir. Immune Defic. Syndr.30, 263–270.

Wahl, L.M., Nowak, M.A., 2000. Adherence and drug resistance: predictions for therapy outcome. Proc.Roy. Soc. Lond. B 267, 835–843.

Wei, X., Ghosh, S.K., Taylor, M.E., Johnson, V.A., Emini, E.A., Deutsch, P., Lifson, J.D., Bonho-effer, S., Nowak, M.A., Hahn, B.H., Saag, M.S., Shaw, G.M., 1995. Viral dynamics in human-immunodeficiency-virus type-1 infection. Nature 373, 117–122.

L. Rong et al.

Wein, L.M., D’Amato, R.M., Perelson, A.S., 1998. Mathematical analysis of antiretroviral therapy aimedat HIV-1 eradication or maintenance of low viral loads. J. Theor. Biol. 192, 81–98.

Wodarz, D., Lloyd, A.L., 2004. Immune responses and the emergence of drug-resistant virus strains invivo. Proc. Roy. Soc. Lond. B 271, 1101–1109.

Wu, H., Huang, Y., Acosta, E.P., Park, J.G., Yu, S., Rosenkranz, S.L., Kuritzkes, D.R., Eron, J.J., Perelson,A.S., Gerber, J.G., 2006. Pharmacodynamics of antiretroviral agents in HIV-1 infected patients: usingviral dynamic models that incorporate drug susceptibility and adherence. J. Pharmacokinet. Pharma-codyn. 33, 399–419.

Wu, H., Huang, Y., Acosta, E.P., Rosenkranz, S.L., Kuritzkes, D.R., Eron, J.J., Perelson, A.S., Gerber,J.G., 2005. Modeling long-term HIV dynamics and antiretroviral response: effects of drug potency,pharmacokinetics, adherence, and drug resistance. J. Acquir. Immune Defic. Syndr. 39, 272–283.

Yuan Chen, H., Di Mascio, M., Perelson, A.S., Ho, D.D., Zhang, L., 2007. Determination of virus burstsize in vivo using a single-cycle SIV in rhesus macaques, submitted.

Zhang, L., Ramratnam, B., Tenner-Racz, K., He, Y., Vesanen, M., Lewin, S., Talal, A., Racz, P., Perelson,A.S., Korber, B.T., Markowitz, M., Ho, D.D., 1999. Quantifying residual HIV-1 replication in patientsreceiving combination antiretroviral therapy. New Engl. J. Med. 340, 1605–1613.

Emergence of HIV-1 Drug Resistance During Antiretroviral ...fengz/pub/bmb07_hiv.pdf · stages of...

Documents

Transcript of Emergence of HIV-1 Drug Resistance During Antiretroviral ...fengz/pub/bmb07_hiv.pdf · stages of...